Maximum ratio transmission

ABSTRACT

An arrangement where a transmitter has a plurality of transmitting antennas that concurrently transmit the same symbol, and where the signal delivered to each transmitting antenna is weighted by a factor that is related to the channel transmission coefficients found between the transmitting antenna and receiving antennas. In the case of a plurality of transmit antennas and one receive antenna, where the channel coefficient between the receive antenna and a transmit antenna i is h i , the weighting factor is h i * divided by a normalizing factor, a, which is 
                 (       ∑     k   =   1     K     ⁢            h   k          2       )       1   /   2       ,         
where K is the number of transmitting antennas. When more than one receiving antenna is employed, the weighting factor is
 
                 1   a     ⁢       (   gH   )     H       ,         
where g=[g 1  . . . g L ), H is a matrix of channel coefficients, and α is a normalizing factor
 
     
       
         
           
             
               
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CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.11/766,853, filed Jun. 22, 2007, now U.S. Pat. No. 7,362,823 which is acontinuation of U.S. patent application Ser. No. 10/963,838 filed onOct. 12, 2004, now U.S. Pat. No. 7,274,752, issued on Sep. 25, 2007,which is a continuation of U.S. patent application Ser. No. 10/177,461filed on Jun. 19, 2002, now U.S. Pat. No. 6,826,236, issued on Nov. 30,2004, which is a continuation of U.S. patent application Ser. No.09/156,066 filed on Sep. 17, 1998, now U.S. Pat. No. 6,459,740, issuedon Oct. 1, 2002, each of which is incorporated by reference in theirentirety herein.

FIELD OF ART

Aspects described herein relate to a system and method for usingtransmit diversity in a wireless communications setting.

BACKGROUND OF THE INVENTION

Wireless communications services are provided in different forms. Forexample, in satellite mobile communications, communications links areprovided by satellite to mobile users. In land mobile communications,communications channels are provided by base stations to the mobileusers. In PCS, communications are carried out in microcell or picocellenvironments, including outdoors and indoors. Regardless the forms theyare in, wireless telecommunication services are provided through radiolinks, where information such as voice and data is transmitted viamodulated electromagnetic waves. That is, regardless of their forms, allwireless communications services are subjected to vagaries of thepropagation environments.

The most adverse propagation effect from which wireless communicationssystems suffer is the multipath fading. Multipath fading, which isusually caused by the destructive superposition of multipath signalsreflected from various types of objects in the propagation environments,creates errors in digital transmission. One of the common methods usedby wireless communications engineers to combat multipath fading is theantenna diversity technique, where two or more antennas at the receiverand/or transmitter are so separated in space or polarization that theirfading envelopes are de-correlated. If the probability of the signal atone antenna being below a certain level is p (the outage probability),then the probability of the signals from L identical antennas all beingbelow that level is p^(L). Thus, since p<1, combining the signals fromseveral antennas reduces the outage probability of the system. Theessential condition for antenna diversity schemes to be effective isthat sufficient de-correlation of the fading envelopes be attained.

A classical combining technique is the maximum-ratio combining (MRC)where the signals from received antenna elements are weighted such thatthe signal-to-noise ratio (SNR) of their sum is maximized. The MRCtechnique has been shown to be optimum if diversity branch signals aremutually uncorrelated and follow a Rayleigh distribution. However, theMRC technique has so far been used exclusively for receivingapplications. As there are more and more emerging wireless services,more and more applications may require diversity at the transmitter orat both transmitter and receiver to combat severe fading effects. As aresult, the interest in transmit diversity has gradually beenintensified. Various transmit diversity techniques have been proposedbut these transmit diversity techniques were built on objectives otherthan to maximize the SNR. Consequently, they are sub-optimum in terms ofSNR performance.

SUMMARY OF THE INVENTION

Improved performance is achieved with an arrangement where thetransmitter has a plurality of transmitting antennas that concurrentlytransmit the same symbol, and where the signal delivered to eachtransmitting antenna is weighted by a factor that is related to thechannel transmission coefficients found between the transmitting antennaand receiving antenna(s). In the case of a plurality of transmitantennas and one receive antenna, where the channel coefficient betweenthe receive antenna and a transmit antenna i is h_(i), the weightingfactor is h_(i)* divided by a normalizing factor, a, which is

${a = ( {\sum\limits_{k = 1}^{K}{h_{k}}^{2}} )^{1/2}},$where K is the number of transmitting antennas. When more than onereceiving antenna is employed, the weighting factor is

${\frac{1}{a}({gH})^{H}},$where g=[g₁ . . . g_(L)], H is a matrix of channel coefficients, and ais a normalizing factor

$( {\sum\limits_{p = 1}^{L}{\sum\limits_{q = 1}^{L}{{\overset{K}{\sum\limits_{k = 1}}{h_{p\; k}h_{qk}^{*}}}}}} )^{1\text{/}2}.$

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an arrangement where there is both transmit andreceive diversity.

FIG. 2 is a flowchart illustrating a routine performed at thetransmitter of FIG. 1.

FIG. 3 is a flowchart illustrating a routine performed at the receiverof FIG. 1.

DETAILED DESCRIPTION

FIG. 1 depicts a system which comprises K antennas for transmission andL antennas for reception. The channel between the transmit antennas andthe receive antennas can be modeled by K×L statistically-independentcoefficients, as show in FIG. 1. It can conveniently be represented inmatrix notation by

$\begin{matrix}{H = {\begin{pmatrix}h_{11} & \cdots & h_{1K} \\\vdots & ⋰ & \vdots \\h_{L\; 1} & \cdots & h_{LK}\end{pmatrix} = \begin{pmatrix}h_{1} \\\vdots \\h_{L}\end{pmatrix}}} & (1)\end{matrix}$where the entry h_(pk) represents the coefficient for the channelbetween transmit antenna k and receiver antenna p. It is assumed thatthe channel coefficients are available to both the transmitter andreceiver through some means, such as through a training session thatemploys pilot signals sent individually through each transmittingantenna (see block 202 of FIG. 2 and block 302 of FIG. 3). Sinceobtaining these coefficients is well known and does not form a part ofthis invention additional exposition of the process of obtaining thecoefficients is deemed not necessary.

The system model shown in FIG. 1 and also in the routines of FIG. 2 andFIG. 3 is a simple baseband representation. The symbol c to betransmitted is weighted with a transmit weighting vector v to form thetransmitted signal vector. The received signal vector, x, is the productof the transmitted signal vector and the channel plus the noise. Thatis,X=Hs+n  (2)where the transmitted signals s is given bys=[s₁ . . . s_(k)]^(T)=c[v₁ . . . v_(k)]^(T),  (3)the channel is represented byH=[h₁ . . . h_(k)],  (4)and the noise signal is expressed asn=[n₁ . . . n_(k)]^(T).  (5)

The received signals are weighted and summed to produce an estimate, ĉ,of the transmitted symbol c.

In accordance with the principles of this invention and as illustratedin block 204 of FIG. 2, the transmit weighting factor, v, is set to

$\begin{matrix}{v = {\frac{1}{a}\begin{bmatrix}h_{1} & \ldots & h_{K}\end{bmatrix}}^{H}} & (6)\end{matrix}$where the superscript H designates the Hermitian operator, and a is anormalization factor given by

$\begin{matrix}{a = ( {\sum\limits_{k = 1}^{K}{h_{k}}^{2}} )^{1/2}} & (7)\end{matrix}$is included in the denominator when it is desired to insure that thetransmitter outputs the same amount of power regardless of the number oftransmitting antennas. Thus, the transmitted signal vector (block 206 ofFIG. 2) is

$\begin{matrix}{s = {{cv} = {\frac{c}{a}\begin{bmatrix}h_{1} & \ldots & h_{K}\end{bmatrix}}^{H}}} & (8)\end{matrix}$and the signal received at one antenna isx=Hs+n=ac+n  (9)from which the symbol can be estimated with the SNR of

$\begin{matrix}{\gamma = {{a^{2}\frac{\sigma_{c}^{2}}{\sigma_{n}^{2}}} = {a^{2}\gamma_{0}}}} & (10)\end{matrix}$where γ₀ denotes the average SNR for the case of a single transmittingantenna (i.e., without diversity). Thus, the gain in the instantaneousSNR is a² when using multiple transmitting antennas rather than a singletransmitting antenna.

The expected value of γ isγ=E[a ²]γ₀ =KE∥h _(k)|²|γ₀  (11)and, hence, the SNR with a K^(th)-order transmitting diversity isexactly the same as that with a K^(th)-order receiving diversity.

When more than one receiving antenna is employed, the weighting factor,v, is

$\begin{matrix}{v = {\frac{1}{a}\lbrack{gH}\rbrack}^{H}} & (12)\end{matrix}$where g=[g₁ . . . g_(L)] (see block 204 of FIG. 2). The transmittedsignal vector is then expressed as

$\begin{matrix}{s = {\frac{c}{a}\lbrack{gh}\rbrack}^{H}} & (13)\end{matrix}$

The normalization factor, a, is |gH|, which yields

$\begin{matrix}{a = ( {\sum\limits_{p = 1}^{L}{\sum\limits_{q = 1}^{L}{g_{p}g_{q}^{*}{\sum\limits_{k = 1}^{K}{h_{p\; k}h_{qk}^{*}}}}}} )^{1/2}} & (14)\end{matrix}$

The received signal vector (block 304 of FIG. 3) is, therefore, given by

$\begin{matrix}{x = {{\frac{c}{a}{H\lbrack{gH}\rbrack}^{H}} + n}} & (15)\end{matrix}$

When the receiver's weighting factor, w, is set to be g (see blocks 306and 308 of FIG. 3), the estimate of the received symbol is given by

$\begin{matrix}{\overset{\_}{c} = {{gx} = {{{\frac{c}{d}{{gH}\lbrack{gh}\rbrack}^{H}} + {gn}} = {{a\; c} + {gn}}}}} & (16)\end{matrix}$with the overall SNR given by

$\begin{matrix}{\gamma = {{\frac{a^{2}}{{gg}^{H}}\gamma_{0}} = \frac{a^{2}\gamma_{0}}{\sum\limits_{p = 1}^{L}{g_{p}}^{2}}}} & (17)\end{matrix}$

From equation (17), it can be observed that the overall SNR is afunction of g. Thus, it is possible to maximize the SNR by choosing theappropriate values of g. Since the h_(qk) terms are assumed to bestatistically identical, the condition that |g₁|=|g₂|= . . . =|g_(L)|has to be satisfied for the maximum value of SNR. Without changing thenature of the problem, one can set |g_(p)|=1 for simplicity. Thereforethe overall SNR is

$\begin{matrix}{\gamma = {\frac{a^{2}}{L}\gamma_{0}}} & (18)\end{matrix}$

To maximize γ is equivalent to maximizing a, which is maximized if

$\begin{matrix}{{g_{p}g_{q}^{*}} = \frac{\sum\limits_{k = 1}^{K}{h_{p\; k}h_{qk}^{*}}}{{\sum\limits_{k = 1}^{K}{h_{p\; k}h_{qk}^{*}}}}} & (19)\end{matrix}$

Therefore,

$\begin{matrix}{a = ( {\sum\limits_{p = 1}^{L}{\sum\limits_{q = 1}^{L}{{\sum\limits_{k = 1}^{K}{h_{p\; k}h_{qk}^{*}}}}}} )^{1/2}} & (20)\end{matrix}$which results in the maximum value of γ. It is clear that the gain inSNR is

$\frac{a^{2}}{L}$when multiple transmitting and receiving antennas are used, as comparedto using a single antenna on the transmitting side or the receivingside.

The vector g is determined (block 306 of FIG. 3) by solving thesimultaneous equations represented by equation (19). For example, ifL=3, equation (19) embodies the following three equations:

$\begin{matrix}{{( {g_{1}g_{2}^{*}} ) = \frac{\sum\limits_{k = 1}^{K}{h_{1\; k}h_{2k}^{*}}}{{\sum\limits_{k = 1}^{K}{h_{1\; k}h_{3k}^{*}}}}},{( {g_{1}g_{3}^{*}} ) = \frac{\sum\limits_{k = 1}^{K}{h_{1\; k}h_{3k}^{*}}}{{\sum\limits_{k = 1}^{K}{h_{1\; k}h_{3k}^{*}}}}},{{{and}\mspace{14mu}( {g_{2}g_{3}^{*}} )} = \frac{\sum\limits_{k = 1}^{K}{h_{2\; k}h_{3k}^{*}}}{{\sum\limits_{k = 1}^{K}{h_{2\; k}h_{3k}^{*}}}}},} & (21)\end{matrix}$

All of the h_(pg) coefficients are known, so the three equations form aset of three equations and three unknowns, allowing a simple derivationof the g₁, g₂, and g₃ coefficients. The corresponding average SNR isgiven by

$\begin{matrix}{\overset{\_}{\gamma} = {{E\lbrack a^{2} \rbrack}\frac{\gamma_{0}}{L}}} & (22)\end{matrix}$where the value of E[a²] depends on the channel characteristics and, ingeneral is bounded byLKE[|h _(k)|² ]≦E[a ² ]≦βL ² KE[|h _(k)|²]  (23)

1. A system comprising: K transmit antennas; a transmitter configured toweight a symbol c by a vector of K weighting factors, to therebygenerate K weighted versions of the symbol c and configured to transmitrespective weighted versions of the symbol c on corresponding antennasof the K transmit antennas, wherein the vector of K weighting factors isproportional to [gH]^(H), where g is a vector having L components andthe superscript H is the Hermitian operator, wherein at least one of Kand L is greater than one.
 2. The system, as recited in claim 1, furthercomprising: L receive antennas; and a receiver configured to receivefrom the L receive antennas, L received versions of the symbol ctransmitted by the K transmit antennas and configured to weight the Lreceived versions of the symbol c by the vector g to generate Lweighted, received versions of the symbol c.
 3. The system, as recitedin claim 2, wherein K is greater than one and L is greater than one. 4.The system, as recited in claim 2, wherein the vector of K weightingfactors is normalized by a normalization factor, a, equal to$( {\sum\limits_{p = 1}^{L}\;{\sum\limits_{q = 1}^{L}\;{g_{p}g_{q}^{*}{\sum\limits_{k = 1}^{K}\;{h_{p\; k}h_{qk}^{*}}}}}} )^{1/2},$wherein h_(pq) is a matrix of coefficients for the channel between the Ktransmit antennas and the L receive antennas.
 5. The system as recitedin claim 4, wherein the normalization factor, a, is$( {\sum\limits_{p = 1}^{L}\;{\sum\limits_{q = 1}^{L}\;{{\sum\limits_{k = 1}^{K}\;{h_{p\; k}h_{qk}^{*}}}}}} )^{1/2}.$6. The system as recited in claim 2, wherein the elements of vector gsatisfy${( {g_{p}g_{q}^{*}} ) = \frac{\sum\limits_{k = 1}^{K}{h_{p\; k}h_{qk}^{*}}}{\;{{\sum\limits_{k = 1}^{K}\;{h_{p\; k}h_{qk}^{*}}}}}},{{\text{where}\text{p}} = 1},2,\ldots\mspace{14mu},\;{{K\mspace{14mu}\text{and}\mspace{14mu} q} = 1},2,\ldots\mspace{14mu},\;{L.}$7. The system as recited in claim 2, wherein the receiver is configuredto sum the L weighted, received versions of the symbol c to thereby forman estimated version, ĉ, of the symbol c.
 8. The system, as recited inclaim 7, wherein an overall signal-to-noise ratio of the estimatedversion ĉ is set to a maximum value.
 9. The system, as recited in claim7, wherein an overall signal-to-noise ratio (SNR) of the estimatedversion ĉ is${\gamma = {\frac{a^{2}}{L}\gamma_{o}}},{{\text{where}\mspace{14mu} a} = ( {\sum\limits_{p = 1}^{L}\;{\sum\limits_{q = 1}^{L}\;{{\sum\limits_{k = 1}^{K}\;{h_{p\; k}h_{qk}^{*}}}}}} )^{1/2}}$and γ_(o) is the average SNR for the case of a single transmittingantenna.
 10. A system comprising: K transmit antennas; L receiveantennas; a transmitter configured to weight a symbol c and configuredto transmit respective weighted versions of the symbol c oncorresponding antennas of the K transmit antennas; and a receiverconfigured to receive from the L receive antennas, L received versionsof the symbol c transmitted by the K transmit antennas and configured toweight the L received versions of the symbol and configured to sum the Lweighted, received versions of the symbol to thereby form an estimatedversion, ĉ, of the symbol c, wherein K is greater than one and L isgreater than one and the overall signal-to-noise ratio (SNR) of theestimated version ĉ is${\gamma = {\frac{a^{2}}{L}\gamma_{o}}},{{\text{where}\mspace{14mu} a} = ( {\sum\limits_{p = 1}^{L}\;{\sum\limits_{q = 1}^{L}\;{{\sum\limits_{k = 1}^{K}\;{h_{p\; k}h_{qk}^{*}}}}}} )^{1/2}},$γ_(o) is the average SNR for the case of a single transmitting antenna,and h_(pq) is the element indexed by p and q of a channel estimatematrix H having K×L elements.
 11. The system, as recited in claim 10,wherein the transmitter weights the symbol c by a vector of K weightingfactors, the vector of K weighting factors being proportional to[gH]^(H), where g is a vector having L components and the superscript His the Hermitian operator.
 12. The system, as recited in claim 10,wherein the receiver weights L received versions of the symbol by thevector g the elements of vector g satisfying${( {g_{p}g_{q}^{*}} ) = \frac{\sum\limits_{k = 1}^{K}{h_{p\; k}h_{qk}^{*}}}{\;{{\sum\limits_{k = 1}^{K}\;{h_{p\; k}h_{qk}^{*}}}}}},{{\text{where}\text{p}} = 1},2,\ldots\mspace{14mu},\;{{K\mspace{14mu}\text{and}\mspace{14mu} q} = 1},2,\ldots\mspace{14mu},\; L,\text{and}$h_(pq) is an element indexed by p and q of a channel estimate matrix Hhaving K×L elements.